J. Math. Comput. Sci. 7 (2017), No. 3, 593-605 ISSN: 1927-5307
INTEGRAL REPRESENTATIONS OF SEVERAL (p,q)-BERNSTEIN
POLYNOMIALS AND THEIR APPLICATIONS
ERKAN AGYUZ∗, MEHMET ACIKGOZ
Department of Mathematics, University of Gaziantep, Gaziantep 27310, Turkey
Copyright c2017 Agyuz and Acikgoz. This is an open access article distributed under the Creative Commons Attribution License, which
permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.
Abstract.The aim of this work is to present some new results of multiplications of(p,q)-Bernstein polynomials and to obtain some new relations of those polynomials with related to(p,q)-Gamma and(p,q)-Beta functions. Keywords: (p,q)-calculus; (p,q)-integral; (p,q)-Bernstein polynomials; (p,q)-Gamma function; (p,q)-Beta function.
2010 AMS Subject Classification:Primary 05A10, 11B65; Secondary 33B65, 11R33.
1. Introduction
Bernstein polynomials [4] have been one of the useful tools for many mathematicians since last century. Getting these tools as afflatus, we have recently seen a lot of important and interesting research studies. We exclusively advert to some of them as follows:
Relations between some special polynomials (Bernoulli and Frobenius-Euler polynomials etc.) and Bernstein polynomials were investigated, very interesting properties were obtained
∗Corresponding author
E-mail address: [email protected]
Received February 28, 2017; Published May 1, 2017
from these researchers [2,5]. Generating function of Bernstein polynomials was obtained. Also recurrence relations and derivative formula for Bernstein polynomials were proved with the aid of generating function [6].
Consider theq- and(p,q)- calculus to obtain many new results. Some new properties were obtained forq- analogue of some special polynomials[20]. Generalized Bernstein polynomials, so calledq-Bernstein polynomials, were defined in [16]. By the motivation from [16], approxi-mation properties for generalize Bernstein polynomials were studied and some estimates on the rate of convergence were given forq-Bernstein polynomials [15]. Results forq-Bernstein poly-nomials were obtained with regard to some special polypoly-nomials [3]. A new generating function for the generalize Bernstein type polynomials was constructed and some basic properties were established by using this new generating function [19]. Acar [1] constructed new modifications of Sz´asz–Mirakyan operators based on (p,q)-integers and derived its new properties. In addi-tion, a new analogue for Bernstein polynomials which is called (p,q)-Bernstein polynomials was defined and approximation properties were studied for these polynomials [14].
This paper is organised as follows:
We will give, respectively, basic definitions and notations which are important and useful for integral representations of Bernstein polynomials in (p,q)-calculus, a definition of (p,q )-Bernstein polynomials introduced by Mursaleen et al.[14], and then extend some well known properties from Bernstein polynomials to(p,q)-Bernstein polynomials and integral representa-tions of(p,q)-Bernstein polynomials and also we will show relations between special functions and these polynomials by using integral representations of (p,q)-Bernstein polynomials. Fi-nally, we will present the conclusions in this paper.
2. Basic Definitions and Notations
In this part, we mention the following definitions and notations that enable us to obtain some results for sequel of this paper.
[n]p,q= p
n−qn
p−q ,0<q<p≤1. (2.1)
If we take p=1, (p,q) analog of n reduces to q analogue of n. Some (p,q)-numbers are determined as follows:
[1]p,q=1 [2]p,q= p+q
[3]p,q= p2+pq+q2
...
Definition 2.2. [9] The(p,q)-Binomial Formula is defined as (a−b)np,q=
n
∑
k=0
n k
p,q
p(n−k)(n−k−1)/2qk(k−1)/2(−1)kan−kbk, (2.2) wherea,b∈R.
Definition 2.3. [9] The(p,q)-Binom coefficients are defined as below:
n
k
p,q
= [n]p,q!
[k]p,q![n−k]p,q!, (2.3) where
[n]p,q!=
1 ,n=0
[n]p,q·[n−1]p,q·...·[1]p,q,n6=0 .
Definition 2.4. [17](p−q)∞
p,qis expressed multiplication of infinite powers as follows:
(p−q)∞p,q=
n
∏
k=0
pn+1−qn+1. (2.4)
Definition 2.5. [18] The(p,q)-derivative operator is determined as (2.5)
Dp,q[f(x)] =
f(px)−f(qx)
(p−q)x (2.5)
Definition 2.6. [9] Let f andg be arbitrary function. The (p,q)- analogue of derivative of product for two functions is defined as
where f :R→Randx∈R.
The definite(p,q)-integral is known as below:
Definition 2.7. [9,18] Suppose that f be an arbitrary function and a be a real number, we construct
Z a
0
f(x)dp,qx= (p−q)a
∞
∑
k=0
qk
pk+1f( qk
pk+1a), q p
<1. (2.7) Whena=1,we see that
Z 1
0
f(x)dp,qx= (p−q)
∞
∑
k=0
qk
pk+1f(
qk
pk+1). For example, taking f(x) =xngives
Z 1
0
xndp,qx= 1 [n+1]p,q.
where0≤a≤b≤∞.
For more information about the applications of(p,q)-integral, see [18].
Now, we present definition of Gamma and Beta functions in(p,q)-calculus which is called post q-calculus. In post q-calculus, these special functions are very important and useful in mathematics, physics and engineering as ordinary sense. Firstly, we begin with definition of (p,q)-Gamma function.
Definition 2.8. [13,17] The(p,q)analogue of Gamma function is determined as (2.9)
Γp,q(x) =
(p−q)∞p,q (px−qx)∞
p,q
(p−q)1−x,0<q< p≤1, (2.9) in which(p,q)-Gamma function satisfies the following conditions:
◦Γp,q(x+1) = [x]p,qΓp,q(x)
◦Γp,q(n+1) = [n]p,q!
wherenis a nonnegative integer.
Definition 2.9. [13] Letmandn∈N. The(p,q)-Beta function is given as
Bp,q(m,n) =
Z 1
0
Remark 2.1.[13](p,q)-Beta function which is defined by (2.10) is not commutatitve. Com-mutative(p,q)-Beta function is determined as below :
e
Bp,q(m,n) =
Z 1
0
pm
(m−1)
2 xm−1(1−qx)n−1
p,q dp,qx.
In postq-calculus, we have important relationships between Gamma and two type Beta func-tions just as ordinary calculus. These relafunc-tions are shown, respectively,
Bp,q(m,n) = p
(n−m)·(2m+n−2)
2 Γp,q(m)·Γp,q(n)
Γp,q(m+n)
e
Bp,q(m,n) = p2mn+m
2+n2−3m−3n+2
2 Γp,q(m)·Γp,q(n)
Γp,q(m+n)
.
3. Main results
In this part, we start by giving the definition of(p,q)-Bernstein operator which is constructed by Mursaleen et. al. in [14].
Definition 3.1. The(p,q)-Bernstein operator is defined by (3.1)
Bn,p,q(f;x) = 1
pn(n2−1) n
∑
k=0
n
k
p,q
pk(k2−1)xk n−k−1
∏
s=0
(ps−qsx)f [k]p,q pk−n[n]
p,q
!
. (3.1)
Definition 3.2.Letkandnbe arbitrary positive integers. The(p,q)-Bernstein polynomial of degreenis given by
Bk,n(x;p,q) =p(k2)−( n 2)
n
k
p,q
xk(1−x)np−,qk, k≤n, (3.2)
where n(n2−1) is shown by n2.
We now give some new corollaries listed below without giving proof because they can be derived by means of the definition of(p,q)-Bernstein polynomials.
Corollary 3.1.(A recursive definition) For 0≤x≤1,we have an equality as below:
Bk,n(x;p,q) = 1−x
q
p
n−k−1!
Corollary 3.2.For 0≤x≤1 and 0<q<p≤1,we obtain an equality as follow:
pn−1 pn−k−1−x qn−k−1+p−k n−1
k
p,q
Bk.n−1(x;p,q) = Bk.n(nx;p,q)
k
p,q
+Bk+1.nn(x;p,q)
k+1
p,q
. (3.4)
Above equation shows that the(p,q)-Bernstein polynomials of degreen−1 is generated by means of(p,q)-Bernstein polynomials of degreen.
Corollary 3.3. (Derivative of(p,q)-Bernstein polynomials) For 0≤x≤1 and 0<q<p≤
1,we obtain an equality as follows:
d
dp,qxBk,n(x;p,q) =
[n]p,q
qkpn−k(qkBk−1,n−1(qx;p,q)−pBk,n−1(qx;p,q))
= nkp,qp k
2
− n2
∑nj−=0k n−jk
p,qp n−k−j
2 q j 2
(−1)j[k+j]p,qxk+j−1.(3.5) By aid of the corollary 3.3, we construct some new corollaries for different type(p,q)-Bernstein polynomials under(p,q)-derivative operator as below:
Corollary 3.4.For 0≤x≤1, 0<q<p≤1 andk1,k2, ...,ks,n∈Nwe obtain an equality as
follows:
d dp,qx B
p,q k1,n(x)∏
s j=2B
p,q kj,n(
q p
(j−1)n−∑lj=−11kl x) ! = ∏ s y=1
n ky
p,q
p
∑sy=1 ky
2
−s n2
+(∑s
m=2((m−1)n−∑ s−1
i=1ki)(n−2ks))
×q(∑sm=2((m−1)n−∑ s−1
i=1ki)ks)×
∑snj=−0(k1+...+ks) sn−(k1+j...+ks)
p,q
×p
sn−(k1+...+ks)−j
2 q j 2
(−1)j[k1+...+ks+j]p,qxk1+...+ks+j−1. Corollary 3.5.For 0≤x≤1, 0<q<p≤1 andki,ni∈N,i=1, ...,s, we have
d
dp,qx Bkp,q
1,n1(x) s
∏
j=2
Bkp,q j,nj(
q
p
∑sy−=11(ny−ky) x) ! = s
∏
y=1
ny ky
p,q
p
∑sy=1 ky
2
− n2y
+∑sj=2
∑ij=−11ni−(s−1)k
(ns−k)
−((∑s
m=2nm−1)−(s−1)k)k
×q((∑sm=2nm−1)−(s−1)k)k
(n1+...+ns)−(k1+...+ks)
∑
j=0
(n1+...+ns)−(k1+...+ks)
j
p,q
×p
(n1+...+ns)−(k1+...+ks)−j
2 q j 2
We are now in a position to state the integral representations of(p,q)-Bernstein polynomials under definite(p,q)-integral over the interval[0,1]. Let us start with the following theorem.
Theorem 3.1.For0<q<p≤1and n−k>0, R1
0Bk,n(qx;p,q)dp,qx p(k2)−(
n 2)qk
=
n
k
p,q n−k
∑
r=0
n−k
r
p,q
(−1)rq(n−2k−r)+rp( r
2) 1
[k+r+1]p,q =p
(n−k)(n+k+1)
2
n k
p,q
[k]p,q[n−k]p,qΓp,q(k)Γp,q(n−k)
Γp,q(n+2)
Proof.Firstly, we consider
Z 1
0
Bk,n(qx;p,q)dp,qx=
Z 1
0
p(k2)−( n 2) n k
p,q
(qx)k(1−qx)np−,qkdp,qx. (3.6)
By using (p,q)-Binomial formula and re-arranging coefficients on right hand side of (3.6), we have
R1
0 Bk,n(qx;p,q)dp,qx p(2k)−(
n 2)qk
=
n
k
p,q
Z 1
0
xk(1−qx)np−,qkdp,qx
=
n k
p,q
Z 1
0
n−k
∑
r=0
n−k
r
p,q
(−1)rq(n−2k−r)+rp( r 2)xk+rd
p,qx
=
n k
p,q n−k
∑
r=0
n−k
r
p,q
(−1)rq(n−2k−r)+rp( r 2)
Z 1
0
xk+rdp,qx
=
n
k
p,q n−k
∑
r=0
n−k
r
p,q
(−1)rq(n−2k−r)+rp( r
2) 1
[k+r+1]p,q. On the other hand, we heuristically know that first equality of above equation has relation both Gamma and Beta functions as follows:
R1
0Bk,n(qx;p,q)dp,qx p(k2)−(
n 2)qk
=
n k
p,q
Z 1
0
xk(1−qx)np−,qkdp,qx
=
n
k
p,q
Bp,q(k+1,n−k+1)
=
n
k
p,q p
(n−k)(n+k+1)
2 Γp,q(k+1)·Γp,q(n−k+1)
Γp,q(n+2)
=
n k
p,q
p(n−k)(n2+k+1)[k]
p,q[n−k]p,q
Γp,q(k)·Γp,q(n−k)
Γp,q(n+2)
.
Some of the integral representations of(p,q)-Bernstein polynomials for special values are as follows:
Z 1
0
B0,1(qx,p,q)dp,qx= p
p+q
Z 1
0
B1,1(qx,p,q)dp,qx= p2q p+q
Z 1
0
B0,2(qx,p,q)dp,qx= (p −1q)2
p2+pq+q2
Z 1
0
B1,2(qx,p,q)dp,qx=
pq2 p2+pq+q2 Z 1
0
B2,2(qx,p,q)dp,qx=q2.
If we takep=1 andqapproaches to 1−, we obtain results in sense of ordinary calculus. We now consider integral representation of multiplication of two(p,q)-Bernstein polynomi-als as below:
Z 1
0
Bk,n(qx;p,q).Bk,m
q
p n−k
qx;p,q !
dp,qx. (3.7)
According to (3.7), we obtain
Theorem 3.2.For 0<q<p≤1 andn+m−2k+1>0,
R1
0Bk,n(qx;p,q).Bk,m
q p
n−k
qx;p,q
dp,qx
p2(k2)−( n 2)−(
m
2)−(n−k)(m−k)q2k
q p
nk−k2
= 2k
∑
r=0 2k
r
p,qp
(r 2)q(
2k−r
2 )+r(−1)r
[n+m+l−2k+1]p,q
= p2k(2n+22m−2k+1)Γp,q(n+m−2k+1)Γp,q(2k+1)
Γp,q(n+m+2)
Proof.Thanks to applying the definition of(p,q)-Bernstein polynomials in (3.7), we get R1
0Bk,n(qx;p,q).Bk,m
q p
n−k
qx;p,q
dp,qx.
p2(k2)−( n 2)−(
m
2)−(n−k)(m−k)q2k
q p
nk−k2
=
n
k
p,q
m
k
p,q
Z 1
0
x2k(1−qx)np−,qk(1−
q
p n−k
qx)mp,−qkdp,qx.
=
n k
p,q
m
k
p,q
Z 1
0
x2k(1−qx)np+,qm−2kdp,qx
=
n k
p,q
m
k
p,q
1 Z
0
xn+m−2k(1−qx)2pk,qdp,qx.
If we use the definition of (p,q)-Binomial formula on right hand side of above equation, we construct
1 R
0
Bk,n(qx;p,q).Bk,m(qpn−kqx;p,q)dp,qx.
p2(k2)−( n 2)−(
m
2)−(n−k)(m−k)q2k
q p
nk−k2
=
n k
p,q
m
k
p,q
2k
∑
r=0
2k r
p,q p(2r)q(
2k−r
2 )+r(−1)r
Z 1
0
xn+m+r−2kdp,qx
=
n k
p,q
m
k
p,q
2k
∑
r=0
(−1)r 2rkp,qp(r2)q( 2k−r
2 )+r
[n+m+r−2k+1]p,q
If we again deal with previous equality, we generate an identity as below:
R1
0 Bk,n(qx;p,q).Bk,m
q p
n−k
qx;p,q
dp,qx
p2(k2)−( n 2)−(
m
2)−(n−k)(m−k)q2k
q p
nk−k2
=
n
k
p,q
m
k
p,q
Bp,q(n+m−2k+1,2k+1)
=
n k
p,q
m
k
p,q p
(n+m−2k)(n+m+2k+1)
2 Γp,q(n+m−2k+1)Γp,q(2k+1)
Γp,q(n+m+2)
.
Thus, the proof is completed.
Theorem 3.3.For 0<q<p≤1 andn+m+s−3k+1>0, R1
0Bk,n(qx;p,q).Bk,m
q p
n−k
qx;p,q
.Bk,s
q p
n+m−2k
qx;p,q
dp,qx
p3(2k)−(( n 2)+(
m 2)+(
s
2))−[(n−k)(m−k)+(n+m−2k)(s−k)]q3k
q p
2nk+mk−3k2
=p
(n+m+s−3k)(n+m+s+3k+1)
2 Γp,q(n+m+s−3k+1)Γp,q(3k+1)
Γp,q(n+m+s+2)
= 3k
∑
r=0 3k
r
p,qp
(r 2)q(
3k−r
2 )+r(−1)r
[n+m+s+r−3k+1]p,q.
Proof.We see that
Z 1
0 "
Bk,n(qx;p,q)×Bk,m
q p
n−k+1
x;p,q
! ×Bk,s
q p
n+m−2k+1
x;p,q
!!#
dp,qx
=
Z 1
0
p(
k 2)−(
n 2) n k
p,q
(qx)k(1−qx)n−kp,q p(k2)−( m 2) m k
p,q
q p
n−k+1
x
!k (1−
q p
n−k+1
x)m−kp,q
×
p(2k)−( s 2) s k
p,q
q p
n+m−2k+1
x
!k (1−
q p
n+m−2k+1
x)s−kp,q.dp,qx
after some basic operations, we obtain: 1
R
0
Bk,n(qx;p,q)×Bk,m
q p
n−k+1 x;p,q
×Bk,s
q p
n+m−2k+1 x;p,q
dp,qx
p3(2k)−(( n 2)+(
m 2)+(
s
2))−[(n−k)(m−k)+(n+m−2k)(s−k)]q3k
q p
2nk+mk−3k2
=
n k
p,q
m
k
p,q
s k
p,q
Z 1
0
x3k(1−qx)np+,qm+s−3kdp,qx
=
n k
p,q
m
k
p,q
s k
p,q
Z 1
0
xn+m+s−3k(1−qx)3pk,qdp,qx.
On the other hand, if we consider(p,q)analogs of Beta and Gamma functions, we have
n k
p,q
m
k
p,q
s k
p,q
Z 1
0
xn+m+s−3k(1−qx)3kp,qdp,qx=
n k
p,q
m
k
p,q
s k
p,q
Bp,q(n+m+s−3k+1,3k+1)
and
Bp,q(n+m+s−3k+1,3k+1) = p
(n+m+s−3k)(n+m+s+3k+1)
2 Γp,q(n+m+s−3k+1)Γp,q(3k+1)
Γp,q(n+m+s+2)
By using the definition of(p,q)-Binomial formula, we obtain
n k
p,q
m
k
p,q
s k
p,q
Z 1
0
xn+m+s−3k(1−qx)3pk,qdp,qx
=
n
k
p,q
m
k
p,q
s
k
p,q
3k
∑
r=0
3k r
p,q p(
r 2)
q(3k2−r)+r
×(−1)r 1 Z
0
xn+m+s−3k+ldp,qx
=
n k
p,q
m
k
p,q
s k
p,q
3k
∑
r=0 3k
r
p,qp (2r)
q(3k2−r)+r(−1)r
[n+m+s+r−3k+1]p,q. Thus, the desired result is obtained.
The following corollary is actually a consequence of a more general previous theorems which we stated and proved in this part.
Corollary 3.6.For 0<q<p≤1,s≥2
R1
0 Bk,n(qx;p,q)∏is=−11Bk,ni+1
q p
∑il=1nl−ik
qx;p,q
dp,qx
ps(k2)−∑ s
i=1(ni2)−(∑ s−1 i=1(∑
i
r=1(nr−ik)(ni+1−k)))qskq p
k∑si=−11(ins−1)−k2(2s)
=
n1
k
p,q
n2
k
p,q
...
ns
k
p,q sk
∑
r=0
sk r
p,qp (r2)
q(sk2−r)+r(−1)r
[n1+n2+...+ns+r−sk+1]p,q
and
R1
0 Bk,n(qx;p,q)∏is=−11Bk,ni+1
q p
∑il=1nl−ik
qx;p,q
dp,qx
ps(k2)−∑is=1(ni2)−(∑ s−1
i=1(∑ir=1(nr−ik)(ni+1−k)))qskq p
k s−1
∑
i=1
(ins−1)−k2(2s)
=
n1
k
p,q
n2
k
p,q
...
ns
k
p,q
Bp,q(sk+1,n1+n2+...+ns−sk+1)
=p[(
n1+n2+...+ns−sk)(n1+n2+...+ns+sk+1)]
2 Γp,q(n1+n2+...+ns−sk+1)Γp,q(sk+1)
Γp,q(n1+n2+...+ns+2)
.
Conflict of Interests
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